Since the root cause of the current upload restriction is not the php.ini (ED already checked that), it is probably an Apache2 configuration, some firewall, or something else that we can't think of right now.
Even though the problem is not in the code of the “Pyra Repository,” we could still develop a workaround for the upload restriction. We could use JavaScript to split the upload into chunks and send them to the server piece by piece using Ajax requests. After the server side has collected all the chunks, the PHP code would concatenate all the chunks again.
I don't want to implement this idea right away. I want to wait a little bit longer to see if ED can fix the problem. However, if he can't fix it, this would probably be a last straw to solve the problem once and for all.